|
|
| Line 1: |
Line 1: |
| [[File:Rado graph.svg|thumb|360px|The Rado graph, as numbered by {{harvtxt|Rado|1964}}.]]
| | The title of the author is Numbers but it's not the most masucline name out there. One of the issues he loves most is ice skating but he is having difficulties to find time for it. Managing people is his occupation. Minnesota is exactly where he's been residing for many years.<br><br>Look at my blog; [http://www.reachoutsociety.info/blog/435 at home std testing] |
| In the [[mathematics|mathematical]] field of [[graph theory]], the '''Rado graph''', also known as the '''random graph''' or the '''Erdős–Rényi graph''', is the unique ([[up to]] [[graph isomorphism|isomorphism]]) [[countable]] graph ''R'' such that for every finite graph ''G'' and every vertex ''v'' of ''G'', every embedding of ''G'' − ''v'' as an [[induced subgraph]] of ''R'' can be extended to an embedding of ''G'' into ''R''. As a result, the Rado graph contains all finite and countably infinite graphs as induced subgraphs.
| |
| | |
| ==History==
| |
| The Rado graph was introduced by {{harvs|first=Richard|last=Rado|authorlink=Richard Rado|txt|year=1964}}, although the symmetry properties of the same graph, constructed in a different way, had already been studied by {{harvtxt|Erdős|Rényi|1963}}.
| |
| | |
| ==Construction via binary numbers==
| |
| {{harvtxt|Rado|1964}} constructs the Rado graph using the [[BIT predicate]] as follows. He identifies the vertices of the graph with the [[natural number]]s 0, 1, 2, ...
| |
| An edge connects vertices ''x'' and ''y'' in the graph (with ''x'' < ''y'') whenever the ''x''th bit of the [[Binary numeral system|binary]] representation of ''y'' is nonzero. Thus, for instance, the neighbors of vertex 0 consist of all odd-numbered vertices, while the neighbors of vertex 1 consist of vertex 0 (the only vertex whose bit in the binary representation of 1 is nonzero) and all vertices with numbers congruent to 2 or 3 modulo 4.
| |
| | |
| ==Finding isomorphic subgraphs==
| |
| The Rado graph satisfies the following extension property: for any finite disjoint sets of vertices ''U'' and ''V'', there exists a vertex ''x'' connected to everything in ''U'', and to nothing in ''V''.
| |
| For instance, let
| |
| :<math>x=2^{1+\max(U\cup V)} + \sum_{u\in U} 2^u.</math>
| |
| Then the nonzero bits in the binary representation of ''x'' cause it to be adjacent to everything in ''U''. However, ''x'' has no nonzero bits in its binary representation corresponding to vertices in ''V'', and ''x'' is so large that the ''x''th bit of every element of ''V'' is zero. Thus, ''x'' is not adjacent to any vertex in ''V''.
| |
| | |
| This idea of finding vertices adjacent to everything in one subset and nonadjacent to everything in a second subset can be used to build up isomorphic copies of any finite or countably infinite graph ''G'', one vertex at a time. For, let ''G''<sub>''i''</sub> denote the subgraph of ''G'' induced by its first ''i'' vertices, and suppose that ''G''<sub>''i''</sub> has already been identified as an induced subgraph of a subset ''S'' of the vertices of the Rado graph. Let ''v'' be the next vertex of ''G'', let ''U'' be the neighbors of ''v'' in ''G''<sub>''i''</sub>, and let ''V'' be the non-neighbors of ''v'' in ''G''<sub>''i''</sub>. If ''x'' is a vertex of the Rado graph that is adjacent to every vertex in ''U'' and nonadjacent to every vertex in ''V'', then ''S'' ∪ {''x''} induces a subgraph isomorphic to ''G''<sub>''i'' + 1</sub>.
| |
| | |
| By induction, starting from the [[empty graph|0-vertex subgraph]], every finite or [[countable set|countably infinite]] graph is an induced subgraph of the Rado graph.
| |
| | |
| ==Uniqueness==
| |
| The Rado graph is, up to [[graph isomorphism]], the only countable graph with the extension property. For, let ''G'' and ''H'' be two graphs with the extension property, let ''G''<sub>''i''</sub> and ''H''<sub>''i''</sub> be isomorphic induced subgraphs of ''G'' and ''H'' respectively, and let ''g''<sub>''i''</sub> and ''h''<sub>''i''</sub> be the first vertices in an enumeration of the vertices of ''G'' and ''H'' respectively that do not belong to ''G''<sub>''i''</sub> and ''H''<sub>''i''</sub>. Then, by applying the extension property twice, one can find isomorphic induced subgraphs ''G''<sub>''i'' + 1</sub> and ''H''<sub>''i'' + 1</sub> that include ''g''<sub>''i''</sub> and ''h''<sub>''i''</sub> together with all the vertices of the previous subgraphs. By repeating this process, one may build up a sequence of isomorphisms between induced subgraphs that eventually includes every vertex in ''G'' and ''H''. Thus, by the [[back-and-forth method]], ''G'' and ''H'' must be isomorphic.<ref name="cam">{{harvtxt|Cameron|2001}}.</ref>
| |
| | |
| By applying the same construction to two isomorphic subgraphs of the Rado graph, it can be shown that the Rado graph is ''ultrahomogeneous'': any isomorphism between any two induced subgraphs of the Rado graph extends to an [[Graph automorphism|automorphism]] of the entire Rado graph.<ref name="cam"/> In particular, there is an automorphism taking any ordered pair of adjacent vertices to any other such ordered pair, so the Rado graph is a [[symmetric graph]].
| |
| | |
| ==Robustness against finite changes==
| |
| If a graph ''G'' is formed from the Rado graph by deleting any finite number of edges or vertices, or adding a finite number of edges, the change does not affect the extension property of the graph: for any pair of sets ''U'' and ''V'' it is still possible to find a vertex in the modified graph that is adjacent to everything in ''U'' and nonadjacent to everything in ''V'', by adding the modified parts of ''G'' to ''V'' and applying the extension property in the unmodified Rado graph. Therefore, any finite modification of this type results in a graph that is isomorphic to the Rado graph.
| |
| | |
| ==Partition property==
| |
| For any partition of the vertices of the Rado graph into two sets ''A'' and ''B'', or more generally for any partition into finitely many subsets, at least one of the subgraphs induced by one of the partition sets is isomorphic to the whole Rado graph. {{harvtxt|Cameron|2001}} gives the following short proof: if none of the parts induces a subgraph isomorphic to the Rado graph, they all fail to have the extension property, and one can find pairs of sets ''U''<sub>''i''</sub> and ''V''<sub>''i''</sub> that cannot be extended within each subgraph. But then, the union of the sets ''U''<sub>''i''</sub> and the union of the sets ''V''<sub>''i''</sub> would form a set that could not be extended in the whole graph, contradicting the Rado graph's extension property. This property of being isomorphic to one of the induced subgraphs of any partition is held by only three countably infinite undirected graphs: the Rado graph, the [[complete graph]], and the [[empty graph]].<ref>{{harvtxt|Cameron|1990}}; {{harvtxt|Diestel|Leader|Scott|Thomassé|2007}}.</ref> {{harvtxt|Bonato|Cameron|Delić|2000}} and {{harvtxt|Diestel|Leader|Scott|Thomassé|2007}} investigate infinite [[directed graph]]s with the same partition property; all are formed by choosing orientations for the edges of the complete graph or the Rado graph.
| |
| | |
| A related result concerns edge partitions instead of vertex partitions: for every partition of the edges of the Rado graph into finitely many sets, there is a subgraph isomorphic to the whole Rado graph that uses at most two of the colors. However, there may not necessarily exist an isomorphic subgraph that uses only one color of edges.<ref>{{harvtxt|Pouzet|Sauer|1996}}.</ref>
| |
| | |
| ==Alternative constructions==
| |
| One may form an infinite graph in the [[Erdős–Rényi model]] by choosing, independently and with probability 1/2 for each pair of vertices, whether to connect the two vertices by an edge. With probability 1 the resulting graph has the extension property, and is therefore isomorphic to the Rado graph. This construction also works if any fixed probability ''p'' not equal to 0 or 1 is used in place of 1/2. This result, shown by [[Paul Erdős]] and [[Alfréd Rényi]] in 1963,<ref>{{harvtxt|Erdős|Rényi|1963}}. Erdős and Rényi use the extension property of graphs formed in this way to show that they have many automorphisms, but do not observe the other properties implied by the extension property. They also observe that the extension property continues to hold for certain sequences of random choices in which different edges have different probabilities of being included.</ref> justifies the [[definite article]] in the common alternative name “the random graph” for the Rado graph: for finite graphs, repeatedly drawing a graph from the Erdős–Rényi model will often lead to different graphs, but for countably infinite graphs the model [[almost always]] produces the same graph. Since one obtains the same random process by inverting all choices, the Rado graph is [[self-complementary graph|self-complementary]].
| |
| | |
| As {{harvtxt|Cameron|2001}} describes, the Rado graph may also be formed by a construction resembling that for [[Paley graph]]s. Take as the vertices of a graph all the [[prime number]]s that are congruent to 1 modulo 4, and connect two vertices by an edge whenever one of the two numbers is a [[quadratic residue]] modulo the other (by [[quadratic reciprocity]] and the restriction of the vertices to primes congruent to 1 mod 4, this is a [[symmetric relation]], so it defines an undirected graph). Then, for any sets ''U'' and ''V'', by the [[Chinese remainder theorem]], the numbers that are quadratic resides modulo every prime in ''U'' and nonresidues modulo every prime in ''V'' form a periodic sequence, so by [[Dirichlet's theorem on arithmetic progressions|Dirichlet's theorem]] on primes in arithmetic progressions this number-theoretic graph has the extension property.
| |
| | |
| ==Related concepts==
| |
| Although the Rado graph is universal for induced subgraphs, it is not universal for [[isometry|isometric embeddings]] of graphs: it has [[diameter (graph theory)|diameter]] two, and so any graph with larger diameter does not embed isometrically into it. {{harvs|last=Moss|year=1989|year2=1991|txt}} has investigated universal graphs for isometric embedding; he finds a family of universal graphs, one for each possible finite graph diameter. The graph in his family with diameter two is the Rado graph.
| |
| | |
| The universality property of the Rado graph can be extended to edge-colored graphs; that is, graphs in which the edges have been assigned to different color classes, but without the usual [[edge coloring]] requirement that each color class form a [[Matching (graph theory)|matching]]. For any finite or countably infinite number of colors χ, there exists a unique countably-infinite χ-edge-colored graph ''G''<sub>χ</sub> such that every partial isomorphism of a χ-edge-colored finite graph can be extended to a full isomorphism. With this notation, the Rado graph is just ''G''<sub>1</sub>. {{harvtxt|Truss|1985}} investigates the automorphism groups of this more general family of graphs.
| |
| | |
| From the [[model theory|model theoretic]] point of view, the Rado graph is an example of a [[saturated model]].
| |
| | |
| {{harvs|last=Shelah|year=1984|year2=1990|txt}} investigates universal graphs with uncountably many vertices.
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| *{{citation
| |
| | doi = 10.4153/CMB-2000-047-6
| |
| | last1 = Bonato | first1 = Anthony
| |
| | last2 = Cameron | first2 = Peter |author2-link = Peter Cameron (mathematician)
| |
| | last3 = Delić | first3 = Dejan
| |
| | mr = 1793941
| |
| | issue = 4
| |
| | journal = Canadian Mathematical Bulletin
| |
| | pages = 397–405
| |
| | title = Tournaments and orders with the pigeonhole property
| |
| | url = http://journals.cms.math.ca/cgi-bin/vault/view/bonato7404
| |
| | volume = 43
| |
| | year = 2000}}.
| |
| *{{citation
| |
| | last = Cameron | first = Peter J.
| |
| | mr = 1066691
| |
| | isbn = 0-521-38836-8
| |
| | location = Cambridge
| |
| | page = viii+160
| |
| | publisher = Cambridge University Press
| |
| | series = London Mathematical Society Lecture Note Series
| |
| | title = Oligomorphic permutation groups
| |
| | volume = 152
| |
| | year = 1990}}.
| |
| *{{citation
| |
| | last = Cameron | first = Peter J.
| |
| | contribution = The random graph
| |
| | mr = 1425227
| |
| | location = Berlin
| |
| | pages = 333–351
| |
| | publisher = Springer
| |
| | series = Algorithms Combin.
| |
| | title = The mathematics of Paul Erdős, II
| |
| | volume = 14
| |
| | year = 1997}}.
| |
| *{{citation
| |
| | last = Cameron | first = Peter J.
| |
| | contribution = The random graph revisited
| |
| | mr = 1905324
| |
| | location = Basel
| |
| | pages = 267–274
| |
| | publisher = Birkhäuser
| |
| | series = Progr. Math.
| |
| | title = European Congress of Mathematics, Vol. I (Barcelona, 2000)
| |
| | volume = 201
| |
| | year = 2001}}.
| |
| *{{citation
| |
| | last = Cameron | first = Peter J. |authorlink = Peter Cameron (mathematician)
| |
| | arxiv = 1301.7544
| |
| | title = The random graph
| |
| | year = 2013
| |
| | accessdate = 2013-06-13}}
| |
| *{{citation
| |
| | last1 = Diestel | first1 = Reinhard
| |
| | last2 = Leader | first2 = Imre
| |
| | last3 = Scott | first3 = Alex
| |
| | last4 = Thomassé | first4 = Stéphan
| |
| | doi = 10.1090/S0002-9947-06-04086-4
| |
| | mr = 2276626
| |
| | issue = 5
| |
| | journal = Transactions of the American Mathematical Society
| |
| | pages = 2395–2405 (electronic)
| |
| | title = Partitions and orientations of the Rado graph
| |
| | volume = 359
| |
| | year = 2007}}.
| |
| *{{citation
| |
| | last1 = Erdős | first1 = P. | author1-link = Paul Erdős
| |
| | last2 = Rényi | first2 = A. | author2-link = Alfréd Rényi
| |
| | doi = 10.1007/BF01895716
| |
| | mr = 0156334
| |
| | journal = Acta Mathematica Academiae Scientiarum Hungaricae
| |
| | pages = 295–315
| |
| | title = Asymmetric graphs
| |
| | volume = 14
| |
| | year = 1963}}.
| |
| *{{citation
| |
| | last = Moss | first = Lawrence S.
| |
| | mr = 1059159
| |
| | issue = 1
| |
| | journal = Polska Akademia Nauk. Fundamenta Mathematicae
| |
| | pages = 25–37
| |
| | title = Existence and nonexistence of universal graphs
| |
| | volume = 133
| |
| | year = 1989}}.
| |
| *{{citation
| |
| | last = Moss | first = Lawrence S.
| |
| | contribution = The universal graphs of fixed finite diameter
| |
| | mr = 1170834
| |
| | location = New York
| |
| | pages = 923–937
| |
| | publisher = Wiley
| |
| | series = Wiley-Intersci. Publ.
| |
| | title = Graph theory, combinatorics, and applications. Vol. 2 (Kalamazoo, MI, 1988)
| |
| | year = 1991}}.
| |
| *{{citation
| |
| | last1 = Pouzet | first1 = Maurice
| |
| | last2 = Sauer | first2 = Norbert
| |
| | doi = 10.1007/BF01271269
| |
| | mr = 1433638
| |
| | issue = 4
| |
| | journal = Combinatorica
| |
| | pages = 505–520
| |
| | title = Edge partitions of the Rado graph
| |
| | volume = 16
| |
| | year = 1996}}.
| |
| *{{citation|first=Richard|last=Rado|authorlink=Richard Rado|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa9/aa9133.pdf|title=Universal graphs and universal functions|journal=Acta Arith.|year=1964|volume=9|pages=331–340}}.
| |
| *{{citation
| |
| | last = Shelah | first = Saharon | author-link = Saharon Shelah
| |
| | doi = 10.1016/0168-0072(84)90042-3
| |
| | mr = 739914
| |
| | issue = 1
| |
| | journal = Annals of Pure and Applied Logic
| |
| | pages = 75–87
| |
| | title = On universal graphs without instances of CH
| |
| | volume = 26
| |
| | year = 1984}}.
| |
| *{{citation
| |
| | last = Shelah | first = Saharon | author-link = Saharon Shelah
| |
| | doi = 10.1007/BF02807219
| |
| | mr = 1057268
| |
| | issue = 1
| |
| | journal = Israel Journal of Mathematics
| |
| | pages = 69–81
| |
| | title = Universal graphs without instances of CH: revisited
| |
| | volume = 70
| |
| | year = 1990}}.
| |
| *{{citation
| |
| | last = Truss | first = J. K.
| |
| | doi = 10.1017/S0305004100063428
| |
| | mr = 795890
| |
| | issue = 2
| |
| | journal = Mathematical Proceedings of the Cambridge Philosophical Society
| |
| | pages = 213–245
| |
| | title = The group of the countable universal graph
| |
| | volume = 98
| |
| | year = 1985}}.
| |
| | |
| [[Category:Individual graphs]]
| |
| [[Category:Random graphs]]
| |
| [[Category:Infinite graphs]]
| |